Z3parafermion in the double charge-Kondo model D. B. Karki1Edouard Boulat2Winston Pouse3 4David Goldhaber-Gordon4 5Andrew K. Mitchell6 7yand Christophe Mora8z

2025-04-29 0 0 858.26KB 16 页 10玖币

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Z3parafermion in the double charge-Kondo model

D. B. Karki,1, ∗Edouard Boulat,2Winston Pouse,3, 4 David

Goldhaber-Gordon,4, 5 Andrew K. Mitchell,6, 7, †and Christophe Mora8, ‡

1Division of Quantum State of Matter, Beijing Academy of Quantum Information Sciences, Beijing 100193, China

2Universit´e de Paris, CNRS, Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, 75013 Paris, France

3Department of Applied Physics, Stanford University, Stanford, CA 94305, USA

4SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA

5Department of Physics, Stanford University, Stanford, CA 94305, USA

6School of Physics, University College Dublin, Belﬁeld, Dublin 4, Ireland

7Centre for Quantum Engineering, Science, and Technology, University College Dublin, Ireland

8Universit´e Paris Cit´e, CNRS, Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, 75013 Paris, France

Quantum impurity models with frustrated Kondo interactions can support quantum critical points

with fractionalized excitations. Recent experiments [W. Pouse et al., Nat. Phys. (2023)] on a circuit

containing two coupled metal-semiconductor islands exhibit transport signatures of such a critical

point. Here we show using bosonization that the double charge-Kondo model describing the device

can be mapped in the Toulouse limit to a sine-Gordon model. Its Bethe-ansatz solution shows

that a Z3parafermion emerges at the critical point, characterized by a fractional 1

2ln(3) residual

entropy, and scattering fractional charges e/3. We also present full numerical renormalization group

calculations for the model and show that the predicted behavior of conductance is consistent with

experimental results.

Quantum impurity models, which feature a few local-

ized, interacting quantum degrees of freedom coupled to

non-interacting conduction electrons, constitute an im-

portant paradigm in the theory of strongly correlated

electron systems [1]. They describe magnetic impurities

embedded in metals or other materials [2,3], and nano-

electronic devices such as semiconductor quantum dots

[4–6] or single-molecule transistors [7,8]. They are also

central to the understanding of bulk correlated materi-

als through dynamical mean ﬁeld theory [9]. General-

ized quantum impurity models host a rich range of com-

plex physics, including various Kondo eﬀects [10–19] and

quantum phase transitions [20–29]. Such models provide

a simple platform to study nontrivial physics which can

be diﬃcult to identify in far more complex bulk materi-

als. Indeed, exact analytical and numerical methods for

quantum impurity models have given deep insights into

strong correlations at the nanoscale [30–34].

The two-channel Kondo (2CK) [10,23] and two-

impurity Kondo (2IK) [21,22] models are classic exam-

ples in which frustrated interactions give rise to non-

Fermi liquid (NFL) physics at quantum critical points

(QCPs) with fractionalized excitations. The seminal

work of Emery and Kivelson (EK) [32] solved the 2CK

model in the Toulouse limit using bosonization tech-

niques, and understood the QCP in terms of a free Ma-

jorana fermion localized on the impurity. In the 2IK

model [22,35–39], a free Majorana arises from the compe-

tition between an RKKY exchange interaction coupling

the impurities, and individual impurity-lead Kondo ef-

fects. In both cases the QCP is characterized by a ﬁnite,

fractional residual impurity entropy of 1

2ln(2) [22,31],

which is a distinctive ﬁngerprint of the free Majorana.

Semiconductor quantum devices [4–6] can constitute

experimental quantum simulators for such impurity mod-

els, with in situ control over parameters allowing cor-

related electron phenomena to be probed with preci-

sion. The distinctive conductance signatures predicted

[24,36,37] for the 2CK model at criticality were since

observed [25,27] (although the 2IK model has never been

realized [40]). More recently, Matveev’s charge-Kondo

paradigm [41,42] has emerged as a viable alternative to

engineer exotic states, with both 2CK [28] and its three-

channel variant [29] being realized experimentally.

Given the intense experimental eﬀorts to demonstrate

the existence of Majoranas in quantum devices [43,44],

and the broader interest in realizing anyons for the pur-

poses of quantum computing [45,46], the Kondo route to

fractionalization has gained traction [47–50]. Experimen-

tal circuit realizations of more complex quantum impu-

rity models oﬀer the tantalizing opportunity to produce

more exotic anyons in tunable nanoelectronics devices.

This can be viewed as part of a wider eﬀort to study

fractionalization in condensed matter systems [51–57].

JLJCJR

𝑵𝑳≡ ⇓ 𝑳⇓𝑹≡ 𝑴 𝑹

𝑳 ↑ 𝑳 ↓ 𝑹 ↓ 𝑹 ↑𝑪 ↑ 𝑪 ↓

⇑𝑹≡ 𝑴 + 𝟏 𝑹

𝑵 + 𝟏 𝑳≡ ⇑ 𝑳

FIG. 1. Schematic of the two-site charge-Kondo circuit de-

scribed by the DCK model. Two hybrid metal-semiconductor

islands are coupled to each other and to their own lead at

QPCs. Macroscopic island charge states mapped to pseu-

dospin degrees of freedom are ﬂipped by tunneling at QPCs.

arXiv:2210.04937v2 [cond-mat.mes-hall] 9 May 2023

However, despite the suggestive fractional entropies in

certain Kondo-type models [29–31,58–61], the explicit

construction of parafermion operators in these systems

has not previously been possible. This is because – un-

like for the simpler case of Majoranas – parafermions can-

not arise in an eﬀective free fermion system. Applying

the EK method yields an irreducibly strongly-interacting

model, which has hitherto hindered ﬁnding exact solu-

tions in which free local parafermions could be identiﬁed.

In this Letter we study the double charge-Kondo

(DCK) model describing a very recent experiment [60]

involving two hybrid metal-semiconductor islands cou-

pled together in series, and each coupled to its own lead,

at quantum point contacts (QPCs) – see Fig. 1. The

DCK model is a variant of the celebrated 2IK model,

but with an inter-island Kondo interaction rather than

an RKKY exchange interaction [21]. At the triple point

in the charge stability diagram of the device, a QCP was

found to arise due to the competition between island-lead

Kondo and inter-island Kondo [60]. Numerical renor-

malization group [33,34,62,63] (NRG) calculations for

the DCK model showed a fractional residual entropy of

2ln(3) at the QCP – suggesting an unusual anyonic state

(and not simply a Majorana). The same critical point

and fractional entropy were identiﬁed analytically near

perfect QPC transmission [64], although no Kondo ef-

fects occur in this limit.

Here we examine the “Kondo” case of weak-to-

intermediate transmission, and apply the EK mapping

[32] in the Toulouse limit. Even though the EK method

yields a highly nontrivial interacting model, we show that

it can nevertheless be solved using Bethe ansatz. Instead

of the free Majorana found by EK for the 2CK model,

we explicitly establish the existence of a Z3parafermion

in the DCK model, and identify it as the source of the

2ln(3) residual entropy. Analytic expressions for conduc-

tance near the QCP are also extracted, and we show that

experimental transport data are consistent with these

predictions. To complete the theoretical description,

we obtain the full temperature dependence of entropy

and conductance via NRG, which does not rely on the

Toulouse approximation.

System and model.– The two-island circuit illustrated

in Fig. 1is described by the DCK model at low tempera-

tures TEC(with ECthe island charging energies) for

weak-to-intermediate QPC transmissions, see Ref. [60]:

HDCK =JLS+

Ls−

L+JRS+

Rs−

R+JCS+

RS−

Ls−

C+ H.c.

−hLSz

L−hRSz

R+ISz

LSz

R+Helec ,

(1)

where Helec =Pα,σ,k kψ†

ασkψασk describes the elec-

tronic reservoirs either side of QPC α=L, C, R. Al-

though the physical electrons are spin-polarized [60],

we label electrons on the lead or island either side of

QPC L, R as σ=↑or ↓, and island electrons to the

left or right of the central QPC Cas σ=↑or ↓– see

Fig. 1. We assume linear dispersion k=vFk, with

momentum k. We then deﬁne pseudospin operators

s−

α=ψ†

α↓(0)ψα↑(0) and s+

α= (s−

α)†, where ψασ (0) is

deﬁned at the QPC position. Conﬁning our attention

to the lowest two macroscopic charge states of each is-

land |n, mi ≡ |niL⊗ |miR, with n=N, N + 1 the num-

ber of electrons on the left island and m=M, M + 1

electrons on the right island, we introduce ‘impurity’

charge pseudospin operators S+

L=Pm|N+1, mihN, m|,

L=Pm1

2[|N+ 1, mihN+ 1, m|−|N, mihN, m|], S+

Pn|n, M + 1ihn, M|,Sz

R=Pn1

2[|n, M + 1ihn, M + 1|−

|n, Mihn, M|] and S−

α= (S+

α)†. The ﬁrst line in Eq. (1)

therefore corresponds to tunneling processes at the three

QPCs (with the tunneling amplitude Jαbeing related to

the transmission ταof QPC α). Gate voltages on the is-

lands control hL,R and allow the charge stability diagram

to be navigated. Iis a capacitive interaction between the

two islands. For JL,C,R =I= 0, the four retained charge

conﬁgurations |n, miare degenerate when hL=hR= 0.

However, a ﬁnite JCand/or Ipartially lifts this degen-

eracy to yield a pair of separated triple points (TPs) in

gate voltage space. As with the experiment [60], here we

focus on the vicinity of the TP at which the charge con-

ﬁgurations |N, Mi/|N+1, Mi/|N, M +1iare degenerate.

We hereafter neglect the term I, since it just renormal-

izes the TP splitting already induced by JC>0 and is

otherwise irrelevant [64]. The rest of this Letter is de-

voted to the nontrivial Kondo competition arising when

the couplings to the leads are switched on, JL,R >0.

QCP.– At the TP, the three ‘impurity’ states (the de-

generate charge conﬁgurations of the two-island struc-

ture) are interconverted by tunneling at the three QPCs:

|N, MiJL

↔ |N+ 1, MiJC

↔ |N, M + 1iJR

↔ |N, Mi

The accompanying conduction electron pseudospin-ﬂip

scattering at each QPC described by the operators s±

L,C,R

in Eq. (1) give rise to competing Kondo eﬀects. Since

island-lead and inter-island Kondo eﬀects cannot be si-

multaneously satisﬁed, a frustration-driven QCP arises

when JL=JR=JC, as reported in Refs. [60,64].

NRG solution.– In Fig. 2we present numerically-exact

results for the DCK model tuned to the TP, obtained

by NRG [33,34,62,63] (see [61] for details). We set

JL=JR≡Jand vary JCin the vicinity of the QCP

arising when JC=J. In panel (a) we show the impurity

contribution to the entropy Simp as a function of tem-

perature T. The critical point JC=J, shown as the red

line, exhibits Kondo ‘overscreening’ to an NFL state on

the scale of TK. The three degenerate charge states give

a high-Tentropy of ln(3), but the entropy is partially

quenched to 1

2ln(3) for TTK. Introducing channel

anisotropy JC6=Jinduces a Fermi liquid (FL) crossover

on the lower scale of T∗, below which the entropy is com-

pletely quenched. The inset shows the extracted power-

law behavior,

T∗/TK∼(|JC−J|/TK)3/2.(2)

(a) (b) (c)

FIG. 2. NRG results at the triple point of the DCK model. (a) Entropy Simp(T) in the vicinity of the critical point, showing

the ﬂow ln(3) →1

2ln(3) on the Kondo scale TK, and subsequently 1

2ln(3) →0 on the Fermi liquid scale T∗. Plotted for

J/D = 0.2 and |JC−J|/D = 10−3,10−4, ..., 10−8(black lines) approaching the critical point JC=J(red line). Dis the

conduction electron bandwidth. Inset shows the power-law behavior Eq. (2). (b) Universal conductance curve as a function of

T/TKat the critical point; (c) Universal Fermi liquid crossover as a function of T /T ∗. Conductance asymptotes are discussed

in the text.

The same form was reported for detuning away from the

TP in Ref. [60]. For |JC−J|  TKwe have good scale

separation T∗TK, such that the crossover to the crit-

ical point is a universal function of the single scaling pa-

rameter T/TK, whereas the crossover away from it is a

universal function of only T /T ∗. This is reﬂected in the

behavior of series conductance, shown in panels (b,c). At

the highest temperatures TTK, Kondo-renormalized

spin-ﬂip scattering gives standard ln−2(T/TK) correc-

tions to conductance; whereas on the lowest temperature

scales TT∗, we observe conventional FL scaling of

conductance ∝(T/T ∗)2. Much more interesting is the

behavior in the vicinity of the critical ﬁxed point [60],

G0−G(T)∼((T/TK)2/3, T TK(3a)

(T/T ∗)−4/3, T T∗(3b)

with G0=e2/3h. Eqs. (2), (3) are also obtained analyt-

ically and discussed in the following.

Bosonization and Toulouse point.– We now turn to

the details of our exact solution. Following EK [32], we

bosonize the conduction electron Hamiltonian Helec and

obtain a simpliﬁed model in the Toulouse limit after ap-

plying a unitary transformation.

As a ﬁrst step we write ψασ =eiφασ /√awith a=

4πvF≡1 and introduce three chiral bosonic ﬁelds δφα≡

(φα↑−φα↓)/√2 for α=L, R, C. The conduction elec-

tron pseudospin operators follow as s−

α=ei√2δφα, and

Helec =vF

4πX

αZdx ∂δφα

∂x 2

.(4)

For hL=hR=I= 0, we can cast the DCK model as,

HDCK =Helec +hJLS+

Lei√2δφL+JRS+

Rei√2δφR

+JCS+

RS−

Lei√2δφC+ H.c.i,(5)

where all ﬁelds are implicitly taken at x= 0. To make

progress, we deform the original DCK model, which fea-

tures only transverse couplings Jα, by adding an Ising

term ¯

HDCK =HDCK +HI. Since pseudospin anisotropy

is RG irrelevant, HIaﬀects only the ﬂow, not the sta-

ble ﬁxed point itself. Therefore the critical ﬁxed point

(and indeed the entire FL crossover in the limit T∗TK

[36,37]) is the same for any choice of HI. We shall exploit

this property to identify an exactly-solvable Toulouse

limit. To do this we eﬀect a change of basis,

δφA= (δφR−δφC−δφL)/√3,

δφB= (δφL+δφR)/√2,

δφD= (δφL−2δφC−δφR)/√6 (6)

and introduce δφ1/2=δφB

√2±δφD

√6. We now choose,

HI=λ[Sz

L∂xδφ1(0) + Sz

R∂xδφ2(0)] ,(7)

and rotate the Hamiltonian into U¯

HDCKU†=Helec +

HEK using the EK unitary transformation [32]

U= exp h−i1

√2{Sz

Lδφ1(0) + Sz

Rδφ2(0)}i.(8)

We then obtain

HEK =hJLS−

L+JRS+

R+JCS+

LS−

Riei√2

3δφA+ H.c.

+¯

λ[Sz

L∂xδφ1(0) + Sz

R∂xδφ2(0)]

(9)

where ¯

λ=λ−1/(4π)2. The Toulouse limit is obtained

by setting ¯

λ= 0, for which the bosonic modes δφB,D

fully decouple and remain free. The symmetric charge

mode δφAthus controls the low-energy behavior follow-

ing Kondo screening. At the QCP with isotropic cou-

plings JL=JR=JC≡J, the model further simpliﬁes,

HEK =J σ ei√2

3δφA+ H.c., σ =



010

001

100

,(10)

where the operator σcircularly permutes the three im-

purity states |N, Mi/|N+ 1, Mi/|N, M + 1i.

Parafermion modes.– In analogy with the descrip-

tion of chiral Potts (clock) models by parafermionic

chains [65], we deﬁne a second operator τ=

diag(1, ω, ω2), with ω=e2iπ/3in the impurity subspace.

The operators [65,66]σand σ0=στ then obey the

parafermionic properties,

σ3=σ03= 1, σσ0=ωσ0σ, (11)

and thereby generalize the Majorana operators to a 3-

dimensional space with circular Z3symmetry.

Importantly, HEK includes only the terms σand σ†,

and not σ0(Eq. (10)). Since σσ†=σ†σ, the parafermion

σcommutes with HEK and remains free. Conversely, σ0

does not commute and it acquires a ﬁnite scaling dimen-

sion.

Sine-Gordon model and Bethe-ansatz solution.– We ro-

tate to the simultaneous eigenbasis of σand σ†and write

HEK =H0⊕H+⊕H−, with

Hr= 2Jcos r2

3δφA+r2π

3!, r = 0,±1.(12)

The DCK model reduces to three decoupled boundary

sine-Gordon models [67–70], related to each other by

aZ3circular shift of the ﬁeld δφA→δφA+ 2π/√6.

They all have the same Bethe-ansatz solution describ-

ing the crossover from high to low energies (the same

crossover as an impurity in a one-dimensional electron

gas with Luttinger parameter K= 1/3 [67]). In partic-

ular, the residual entropy is predicted [71,72] to decrease

by ∆S=1

2ln(3) along the crossover. For the DCK model

we therefore expect a crossover in the impurity entropy

from ln(3) to 1

2ln(3), as conﬁrmed by the NRG results

in Fig. 2(a). The parafermions σand σ0generate the

threefold charge subspace. Since σ0is screened but σre-

mains free, it simply halves the residual entropy. The

same residual entropy was found in the quasi-ballistic

limit [64].

Conductance at the critical point.– The linear conduc-

tance between left and right leads is obtained from the

Kubo formula G=−2πlimω→0[Im K(ω)/ω], with K(ω)

the Fourier transform of the retarded current-current cor-

relator K(t) = −iθ(t)h[I(t), I(0)]i. Following the above

mapping, I=−e

2πq2

3∂tΘA, where ΘAis the ﬁeld con-

jugate to δφA. Since δφAis pinned at the critical ﬁxed

point, ΘAis free, and so hΘA(t)ΘA(0)i∼−ln tat T= 0.

This yields G=G0=e2/3h: out of the three ﬁelds, only

φAappears in Eq. (9), thus only ΘAtransports electrons,

yielding 1/3 of a perfect conductance.

Conductance scaling in the Kondo regime, TTK.–

We now turn to the leading ﬁnite-temperature correc-

tions to the T= 0 conductance at the critical point. To

do this, we must perturb away from the exactly solvable

EK point by reintroducing ﬁnite ¯

λ. This is because the

RG ﬂow to the critical ﬁxed point is aﬀected by ¯

λ. The

leading irrelevant operator (LIO) at the QCP is given

by OLIO =¯

λ[Sz

L∂xδφ1(0) + Sz

R∂xδφ2(0)]. As we show in

[61], the operators ∂xδφ1,2(0) both have scaling dimen-

sion 1 and Sz

L,Rhave scaling dimension 1

3. This yields

∆LIO = 4/3, and therefore allows us to identify the lead-

ing correction to conductance (arising at order ¯

λ2) as

δG ∼(T/TK)2(∆LIO−1) [61], which reproduces Eq. (3a).

FL crossover.– The QCP is destabilized by gate volt-

age detuning away from the TP (appearing as pseudo-

Zeeman ﬁelds hL,R in the DCK model), or by channel

anisotropy δJ. The resulting FL crossover is controlled

by the FL scale T∗. Assuming T∗TK, we may again

utilize the Toulouse limit and set ¯

λ= 0 to analyze the FL

crossover, since any ﬁnite ¯

λscales to zero anyway under

RG for TTK. Both perturbations hL,R and δJ have

the eﬀect of coupling the otherwise independent sectors

of HEK given by Eq. (12). We focus here on ﬁnite hRfor

simplicity. From Eq. (1), hRcouples to Sz

R, which in the

rotated basis is given by Sz

R=1

3(ωτ +ω∗τ†). Analyzing

its action at the QCP [61], we identify τ=e−i√2/3ΘA,

where this operator circularly permutes the sectors r

in Eq. (12). Sz

Rthus inherits the RG-relevant scaling

dimension ∆R= 1/3 of τ, such that ﬁnite hRgen-

erates a FL scale T∗∼h1/(1−∆R)

R. Since Sz

Land δJ

have the same scaling dimension ∆R, in general we have

T∗∼(h3/2

L, h3/2

R, δJ3/2) [61], which reduces to Eq. (2) in

the case of pure channel anisotropy. The leading correc-

tion in T/T ∗to the QCP conductance G0then follows

as δG ∼(T/T ∗)2(∆R−1), yielding Eq. (3b). Addition-

ally, the free parafermion at the QCP is shown by noise

calculation [61] to scatter fractional charges e∗=e/3.

Comparison with experiment.– Finally we turn to

the implications of our results for the experiments of

Ref. [60]. Although the experimental results were ob-

tained at large transmission τ,τC, we expect the uni-

versal low-temperature behavior near the QCP to be

the same as that discussed above for the Kondo limit

[61,64]. Since the maximum conductance measured is

slightly lower than the predicted value G0=e2/3h, we

infer that the quantum critical state is not fully devel-

oped at experimental base temperatures. Detuning away

from the TP by varying the island gate voltages Ugener-

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Z3parafermioninthedoublecharge-KondomodelD.B.Karki,1,EdouardBoulat,2WinstonPouse,3,4DavidGoldhaber-Gordon,4,5AndrewK.Mitchell,6,7,yandChristopheMora8,z1DivisionofQuantumStateofMatter,BeijingAcademyofQuantumInformationSciences,Beijing100193,China2UniversitedeParis,CNRS,LaboratoireMateriauxetPheno...

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Z3parafermion in the double charge-Kondo model D. B. Karki1Edouard Boulat2Winston Pouse3 4David Goldhaber-Gordon4 5Andrew K. Mitchell6 7yand Christophe Mora8z

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